We present a two-step variant of the "successive band reduction" paradigm for the tridiagonalization of symmetric matrices. We first reduce a full matrix to narrow-banded form, and from there to tridiagonal form. The first step allows easy exploitation of block orthogonal transformations. In the second step, we employ a new blocked version of a banded matrix tridiagonalization algorithm by Lang (1993). In particular, we are able to express the update of the orthogonal transformation matrix in terms of block transformations, which leads to an algorithm that is almost entirely based on BLAS-3 kernels, and has greatly improved data movement and communication characteristics. We also present some performance results on the Intel Touchstone Delta prototype and the IBM SP/1.
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